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# Exercise 98 - Chapter 1

edited June 2018

To be sure that $$g$$ really is right adjoint to $$f$$ in Example 1.97 [below], there are twelve things to check; do so. That is, for every $$p \in P$$ and $$q \in Q$$, check that $$f(p) \le q \text{ iff } p \le g(q)$$.

Let $$g$$ be the map that preserves labels, and let $$f$$ be the map that preserves labels as far as possible but sends $$f(3.9) = 4$$.

$$\begin{array}{c|c|c|c} p&f(p)&g(f(p))&p \leq g(f(p))\\ \hline 1 &1&1&T\\ 2 &2&2&T\\ 3.9&4&4&T\\ 4 &4&4&T\\ \end{array}$$ and
$$\begin{array}{c|c|c|c} q&g(q)&f(g(q))&f(g(q)) \leq q\\ \hline 1&1&1&T\\ 2&2&2&T\\ 4&4&4&T\\ \end{array}$$
Comment Source:Using Proposition 1.93, we can check just 7 things: $\begin{array}{c|c|c|c} p&f(p)&g(f(p))&p \leq g(f(p))\\\\ \hline 1 &1&1&T\\\\ 2 &2&2&T\\\\ 3.9&4&4&T\\\\ 4 &4&4&T\\\\ \end{array}$ and $\begin{array}{c|c|c|c} q&g(q)&f(g(q))&f(g(q)) \leq q\\\\ \hline 1&1&1&T\\\\ 2&2&2&T\\\\ 4&4&4&T\\\\ \end{array}$